# Properties

 Label 882.3.c.d Level $882$ Weight $3$ Character orbit 882.c Analytic conductor $24.033$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 882.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.0327593166$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.2048.2 Defining polynomial: $$x^{4} + 4 x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + 2 q^{4} + ( \beta_{1} + 2 \beta_{3} ) q^{5} + 2 \beta_{2} q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} + 2 q^{4} + ( \beta_{1} + 2 \beta_{3} ) q^{5} + 2 \beta_{2} q^{8} + ( \beta_{1} + 3 \beta_{3} ) q^{10} + 2 \beta_{2} q^{11} + ( 5 \beta_{1} + 8 \beta_{3} ) q^{13} + 4 q^{16} + ( -6 \beta_{1} + 9 \beta_{3} ) q^{17} + ( 6 \beta_{1} - 10 \beta_{3} ) q^{19} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{20} + 4 q^{22} + ( -28 - 6 \beta_{2} ) q^{23} + ( 15 - 7 \beta_{2} ) q^{25} + ( 3 \beta_{1} + 13 \beta_{3} ) q^{26} + ( -28 + 11 \beta_{2} ) q^{29} + ( -14 \beta_{1} + 14 \beta_{3} ) q^{31} + 4 \beta_{2} q^{32} + ( 15 \beta_{1} + 3 \beta_{3} ) q^{34} + ( 16 + 21 \beta_{2} ) q^{37} + ( -16 \beta_{1} - 4 \beta_{3} ) q^{38} + ( 2 \beta_{1} + 6 \beta_{3} ) q^{40} + ( 35 \beta_{1} + 14 \beta_{3} ) q^{41} + ( -32 - 28 \beta_{2} ) q^{43} + 4 \beta_{2} q^{44} + ( -12 - 28 \beta_{2} ) q^{46} + ( -10 \beta_{1} + 50 \beta_{3} ) q^{47} + ( -14 + 15 \beta_{2} ) q^{50} + ( 10 \beta_{1} + 16 \beta_{3} ) q^{52} + ( 42 + 44 \beta_{2} ) q^{53} + ( 2 \beta_{1} + 6 \beta_{3} ) q^{55} + ( 22 - 28 \beta_{2} ) q^{58} + ( -46 \beta_{1} + 6 \beta_{3} ) q^{59} + ( 37 \beta_{1} - 8 \beta_{3} ) q^{61} + 28 \beta_{1} q^{62} + 8 q^{64} + ( -42 - 29 \beta_{2} ) q^{65} + 4 q^{67} + ( -12 \beta_{1} + 18 \beta_{3} ) q^{68} + ( -112 + 14 \beta_{2} ) q^{71} + ( -32 \beta_{1} - 33 \beta_{3} ) q^{73} + ( 42 + 16 \beta_{2} ) q^{74} + ( 12 \beta_{1} - 20 \beta_{3} ) q^{76} + ( -36 + 28 \beta_{2} ) q^{79} + ( 4 \beta_{1} + 8 \beta_{3} ) q^{80} + ( -21 \beta_{1} + 49 \beta_{3} ) q^{82} + ( 76 \beta_{1} + 12 \beta_{3} ) q^{83} + ( -24 - 21 \beta_{2} ) q^{85} + ( -56 - 32 \beta_{2} ) q^{86} + 8 q^{88} + ( -30 \beta_{1} + 73 \beta_{3} ) q^{89} + ( -56 - 12 \beta_{2} ) q^{92} + ( 60 \beta_{1} + 40 \beta_{3} ) q^{94} + ( 28 + 24 \beta_{2} ) q^{95} + ( -64 \beta_{1} + 39 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{4} + O(q^{10})$$ $$4q + 8q^{4} + 16q^{16} + 16q^{22} - 112q^{23} + 60q^{25} - 112q^{29} + 64q^{37} - 128q^{43} - 48q^{46} - 56q^{50} + 168q^{53} + 88q^{58} + 32q^{64} - 168q^{65} + 16q^{67} - 448q^{71} + 168q^{74} - 144q^{79} - 96q^{85} - 224q^{86} + 32q^{88} - 224q^{92} + 112q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4 x^{2} + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 3 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$785$$ $$\chi(n)$$ $$-1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
685.1
 − 1.84776i 1.84776i − 0.765367i 0.765367i
−1.41421 0 2.00000 0.317025i 0 0 −2.82843 0 0.448342i
685.2 −1.41421 0 2.00000 0.317025i 0 0 −2.82843 0 0.448342i
685.3 1.41421 0 2.00000 4.46088i 0 0 2.82843 0 6.30864i
685.4 1.41421 0 2.00000 4.46088i 0 0 2.82843 0 6.30864i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.3.c.d 4
3.b odd 2 1 882.3.c.e yes 4
7.b odd 2 1 inner 882.3.c.d 4
7.c even 3 2 882.3.n.h 8
7.d odd 6 2 882.3.n.h 8
21.c even 2 1 882.3.c.e yes 4
21.g even 6 2 882.3.n.g 8
21.h odd 6 2 882.3.n.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.3.c.d 4 1.a even 1 1 trivial
882.3.c.d 4 7.b odd 2 1 inner
882.3.c.e yes 4 3.b odd 2 1
882.3.c.e yes 4 21.c even 2 1
882.3.n.g 8 21.g even 6 2
882.3.n.g 8 21.h odd 6 2
882.3.n.h 8 7.c even 3 2
882.3.n.h 8 7.d odd 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{4} + 20 T_{5}^{2} + 2$$ $$T_{23}^{2} + 56 T_{23} + 712$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$2 + 20 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( -8 + T^{2} )^{2}$$
$13$ $$3362 + 356 T^{2} + T^{4}$$
$17$ $$46818 + 468 T^{2} + T^{4}$$
$19$ $$67712 + 544 T^{2} + T^{4}$$
$23$ $$( 712 + 56 T + T^{2} )^{2}$$
$29$ $$( 542 + 56 T + T^{2} )^{2}$$
$31$ $$307328 + 1568 T^{2} + T^{4}$$
$37$ $$( -626 - 32 T + T^{2} )^{2}$$
$41$ $$8072162 + 5684 T^{2} + T^{4}$$
$43$ $$( -544 + 64 T + T^{2} )^{2}$$
$47$ $$23120000 + 10400 T^{2} + T^{4}$$
$53$ $$( -2108 - 84 T + T^{2} )^{2}$$
$59$ $$4669568 + 8608 T^{2} + T^{4}$$
$61$ $$1016738 + 5732 T^{2} + T^{4}$$
$67$ $$( -4 + T )^{4}$$
$71$ $$( 12152 + 224 T + T^{2} )^{2}$$
$73$ $$8380418 + 8452 T^{2} + T^{4}$$
$79$ $$( -272 + 72 T + T^{2} )^{2}$$
$83$ $$111183872 + 23680 T^{2} + T^{4}$$
$89$ $$155196962 + 24916 T^{2} + T^{4}$$
$97$ $$11683778 + 22468 T^{2} + T^{4}$$
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